By the structure of parabolic subgroups [1]}, the minimal parabolic subgroup \(P\le G\)  is equal to the semi-direct product \(_G(T)\ltimes {u}{P}\) , where \(\)  is a maximal \(k\) -split torus, and \(_G(T)\)  is the centralizer of \(T\)  in \(G\) .
Denote by \(\)  the solvable \(k\) -algebraic group \( \ltimes {u}{}\) .
We claim that \(Z(A)<Z(P)\) .
Indeed, for \(a\in Z(A)\) , since \(T\subset A\) , clearly \(a\in _G(T)\)  hence \(a\in T\) , so \([a,_G(T)]=\lbrace e\rbrace \)  and thus \(a\in Z(P)\) .
Following Lemma REF , \(Z(P)=Z(G)\) , and \(Z(G)\)  is compact, therefore \(Z(A)\)  is compact.
Hence, by Proposition REF  the group \(A\)  is minimal.
Moreover, let \(Z_0\le Z(P)\)  be a \(k\) -algebraic subgroup. As \(Z(P)=Z(G)\) , the group \(Z_0\)  is central in \(G\)  and is therefore a central algebraic torus.
The group \(A^{\prime }=TZ_0\ltimes {u}{G}\)  is equal to the \(k\) -points of the \(k\) -algebraic group \(_0\) .
Since \(T\subset A^{\prime }\)  for any \(a^{\prime }\in Z(A^{\prime })\) , \(a^{\prime }\in _G(T)\)  hence \(a\in TZ_0\)  and therefore commutes with \(_G(T)\)  as well, making \(a\in Z(P)\) . Hence \(Z(A)\)  is compact, and by Proposition REF  the group \(A^{\prime }/Z_0\)  is minimal.
Following [2]}, the anisotropic group \(_G(T)/T\)  is compact. Therefore the solvable subgroup \(A\le P\)  is co-compact, and in particular, \(A^{\prime }\)  is co-compact in \(P\) .
Hence \(A\) , ( \(A^{\prime }/Z_0\) ), is a minimal co-compact subgroup of \(P\) , (\(P/Z_0\) ) thus by Lemma REF , \(P\)  (\(P/Z_0\) ), is minimal.
